Solve for x: log₈ x + log₄ x + log₂ x = 11. Use base conversion (let y = log₂ x) to reduce to a linear equation.

Introduction / Context:The equation mixes logarithms with bases 8, 4, and 2 applied to the same x. Converting all to base 2 simplifies the problem to a single-variable linear equation.

Given Data / Assumptions:

• log₈ x + log₄ x + log₂ x = 11
• x > 0

Concept / Approach:Let y = log₂ x, so x = 2^y. Then log₄ x = y/2 (since 4 = 2^2) and log₈ x = y/3 (since 8 = 2^3). Substitute and solve for y, then recover x = 2^y.

Step-by-Step Solution:

Verification / Alternative check:Compute each term: log₂ 64 = 6, log₄ 64 = 3, log₈ 64 = 2; sum 6 + 3 + 2 = 11, confirming the solution.

Why Other Options Are Wrong:2, 4, and 8 yield sums far smaller than 11; only 64 produces the required sum.

Common Pitfalls:Forgetting the base-change scaling (dividing by 2 or 3) or mis-adding fractions can produce a wrong y value. Carefully use common denominators.

Final Answer:64